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8.2: Three cases for λ

https://math.libretexts.org/Bookshelves/Differential_Equations/Partial_Differential_Equations_(Walet)/08%3A_Separation_of_Variables_in_Polar_Coordinates/8.02%3A_Three_cases_for_%CE%BB

Now let me look at the solution of the

$R$ equation for each of the two cases (they can be treated as one),

$\rho^2 R''(\rho) + \rho R'(\rho) -n^2R(\rho)=0. \nonumber$ Let us attempt a power-seri...Now let me look at the solution of the

$R$ equation for each of the two cases (they can be treated as one),

$\rho^2 R''(\rho) + \rho R'(\rho) -n^2R(\rho)=0. \nonumber$ Let us attempt a power-series solution (this method will be discussed in great detail in a future lecture)

$R(\rho) =\rho^\alpha. \nonumber$ We find the equation

$\rho^\alpha[\alpha(\alpha-1)+\alpha^2-n^2]= \rho^\alpha[\alpha^2-n^2]=0 \nonumber$ If

$n\neq 0$ we thus have two independent solutions (as should be) \[R_n…

3: Boundary and Initial Conditions

https://math.libretexts.org/Bookshelves/Differential_Equations/Partial_Differential_Equations_(Walet)/03%3A_Boundary_and_Initial_Conditions

As you all know, solutions to ordinary differential equations are usually not unique (integration constants appear in many places). This is of course equally a problem for PDE’s. PDE’s are usually spe...As you all know, solutions to ordinary differential equations are usually not unique (integration constants appear in many places). This is of course equally a problem for PDE’s. PDE’s are usually specified through a set of boundary or initial conditions. A boundary condition expresses the behavior of a function on the boundary (border) of its area of definition. An initial condition is like a boundary condition, but then for the time-direction.

10.7: Our Initial Problem and Bessel Functions

https://math.libretexts.org/Bookshelves/Differential_Equations/Partial_Differential_Equations_(Walet)/10%3A_Bessel_Functions_and_Two-Dimensional_Problems/10.07%3A_Our_Initial_Problem_and_Bessel_Functions

We started the discussion from the problem of the temperature on a circular disk, solved in polar coordinates, Since the initial conditions do not depend on

$\phi$ , we expect the solution to be radi...We started the discussion from the problem of the temperature on a circular disk, solved in polar coordinates, Since the initial conditions do not depend on

$\phi$ , we expect the solution to be radially symmetric as well.

10: Bessel Functions and Two-Dimensional Problems

https://math.libretexts.org/Bookshelves/Differential_Equations/Partial_Differential_Equations_(Walet)/10%3A_Bessel_Functions_and_Two-Dimensional_Problems

6: D’Alembert’s Solution to the Wave Equation

https://math.libretexts.org/Bookshelves/Differential_Equations/Partial_Differential_Equations_(Walet)/06%3A_DAlemberts_Solution_to_the_Wave_Equation

It is usually not useful to study the general solution of a partial differential equation. As any such sweeping statement it needs to be qualified, since there are some exceptions. One of these is the...It is usually not useful to study the general solution of a partial differential equation. As any such sweeping statement it needs to be qualified, since there are some exceptions. One of these is the one-dimensional wave equation which has a general solution, due to the French mathematician d’Alembert.

4.2: Introduction to Fourier Series

https://math.libretexts.org/Bookshelves/Differential_Equations/Partial_Differential_Equations_(Walet)/04%3A_Fourier_Series/4.02%3A_Introduction_to_Fourier_Series

Rather than Taylor series, that are supposed to work for “any” function, we shall study periodic functions. For periodic functions the French mathematician introduced a series in terms of sines and co...Rather than Taylor series, that are supposed to work for “any” function, we shall study periodic functions. For periodic functions the French mathematician introduced a series in terms of sines and cosines, We shall study how and when a function can be described by a Fourier series. One of the very important differences with Taylor series is that they can be used to approximate non-continuous functions as well as continuous ones.

4.3: Periodic Functions

https://math.libretexts.org/Bookshelves/Differential_Equations/Partial_Differential_Equations_(Walet)/04%3A_Fourier_Series/4.03%3A_Periodic_Functions

A function is called periodic with period

$p$ if

$f(x+p)=f(x)$ , for all

$x$ , even if

$f$ is not defined everywhere. A simple example is the function

$f(x)=\sin(bx)$ which is periodic with pe...A function is called periodic with period

$p$ if

$f(x+p)=f(x)$ , for all

$x$ , even if

$f$ is not defined everywhere. A simple example is the function

$f(x)=\sin(bx)$ which is periodic with period

$(2π)∕b$ . In general a function with period

$p$ is periodic with period 2p 3p …. This can easily be seen using the definition of periodicity, which subtracts p from the argument The smallest positive value of p for which f is periodic is called the (primitive) period of f.

11: Separation of Variables in Three Dimensions

https://math.libretexts.org/Bookshelves/Differential_Equations/Partial_Differential_Equations_(Walet)/11%3A_Separation_of_Variables_in_Three_Dimensions

We have up to now concentrated on 2D problems, but a lot of physics is three dimensional, and often we have spherical symmetry – that means symmetry for rotation over any angle.

4.7: Convergence of Fourier series

https://math.libretexts.org/Bookshelves/Differential_Equations/Partial_Differential_Equations_(Walet)/04%3A_Fourier_Series/4.07%3A_Convergence_of_Fourier_series

It is not very hard to find the relevant Fourier series, \[\begin{aligned} f(x) & = & -\frac{4}{\pi} \sum_{m=0}^\infty \frac{1}{2m+1} \sin (2m+1) x,\\ g(x) & = & \frac{4}{\pi} \sum_{m=0}^\infty \frac{...It is not very hard to find the relevant Fourier series,

$\begin{aligned} f(x) & = & -\frac{4}{\pi} \sum_{m=0}^\infty \frac{1}{2m+1} \sin (2m+1) x,\\ g(x) & = & \frac{4}{\pi} \sum_{m=0}^\infty \frac{1}{(2m+1)^2} \cos (2m+1) x.\end{aligned} \nonumber$ Let us compare the partial sums, where we let the sum in the Fourier series run from

$m=0$ to

$m=M$ instead of

$m=0\ldots\infty$ .

7.2: Spherical Coordinates

https://math.libretexts.org/Bookshelves/Differential_Equations/Partial_Differential_Equations_(Walet)/07%3A_Polar_and_Spherical_Coordinate_Systems/7.02%3A_Spherical_Coordinates

The spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers. Integrating requires a volume element.

4.1: Taylor Series

https://math.libretexts.org/Bookshelves/Differential_Equations/Partial_Differential_Equations_(Walet)/04%3A_Fourier_Series/4.01%3A_Taylor_Series

\[\begin{aligned} &&\qquad&\cos(0) &= 1,\nonumber\\ \cos'(x) &= -\sin(x),&&\cos'(0)&=0,\nonumber\\ \cos^{(2)}(x) &= -\cos(x),&&\cos^{(2)}(0)&=-1,\\ \cos^{(3)}(x) &= \sin(x),&&\cos^{(3)}(0)&=0,\nonumbe...

$\begin{aligned} &&\qquad&\cos(0) &= 1,\nonumber\\ \cos'(x) &= -\sin(x),&&\cos'(0)&=0,\nonumber\\ \cos^{(2)}(x) &= -\cos(x),&&\cos^{(2)}(0)&=-1,\\ \cos^{(3)}(x) &= \sin(x),&&\cos^{(3)}(0)&=0,\nonumber\\ \cos^{(4)}(x) &= \cos(x),&&\cos^{(4)}(0)&=1.\end{aligned} \nonumber$

$\cos x = \sum_{m=0}^\infty \frac{(-1)^m}{(2m)!} x^{2m}, \nonumber$ Show that

$\sin x = \sum_{m=0}^\infty \frac{(-1)^m}{(2m+1)!} x^{2m+1}. \nonumber$

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